Προσαρτημένη Μήτρα
Προσηρτημένη Μήτρα Adjoint Matrix, Μιγαδικά συζυγής Μαθηματική Μήτρα ---- Μηδενική Μήτρα Μοναδιαία Μήτρα Διαγώνια Μήτρα Τριγωνική Μήτρα Τριδιαγώνια Μήτρα Συζυγής Μήτρα Ανάστροφη Μήτρα Συμμετρική Μήτρα Αντισυμμετρική Μήτρα Προσαρτημένη Μήτρα Ερμιτιανή Μήτρα Ανθερμιτιανή Μήτρα Κανονική Μήτρα Αντίστροφη Μήτρα Μοναδιακή Μήτρα Ορθογώνια Μήτρα ---- Ίχνος Μήτρας Ορίζουσα Μήτρας ---- Μητραϊκή Πρόσθεση Μητραϊκός Πολλαπλασιασμός Μητραϊκή Σύνθεση Μητραϊκή Αναπαράσταση ]] Μιγαδική Συζυγής Μήτρα Ανάστροφη Μήτρα Ερμιτιανή Μήτρα]] - Ένα Μαθηματικό Μέγεθος. Ετυμολογία Η ονομασία "Μήτρα" σχετίζεται ετυμολογικά με την λέξη "μήτηρ". Εισαγωγή In mathematics, the conjugate transpose or Hermitian transpose of an -by- matrix with complex entries is the -by- matrix obtained from by taking the transpose and then taking the complex conjugate of each entry. (The complex conjugate of , where and are reals, is .) The conjugate transpose is formally defined by : (\boldsymbol{A}^*)_{ij} = \overline{\boldsymbol{A}_{ji}} where the subscripts denote the ( , )-th entry, for ≤ }} and ≤ }}, and the overbar denotes a scalar complex conjugate. This definition can also be written as : \boldsymbol{A}^* = (\overline{\boldsymbol{A}})^\mathrm{T} = \overline{\boldsymbol{A}^\mathrm{T}} where \boldsymbol{A}^\mathrm{T} denotes the transpose and \overline{\boldsymbol{A}} denotes the matrix with complex conjugated entries. Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. The conjugate transpose of a matrix can be denoted by any of these symbols: * \boldsymbol{A}^* or \boldsymbol{A}^\mathrm{H} , commonly used in linear algebra * \boldsymbol{A}^\dagger (sometimes pronounced as " dagger"), universally used in quantum mechanics * \boldsymbol{A}^+ , although this symbol is more commonly used for the Moore–Penrose pseudoinverse In some contexts, \boldsymbol{A}^* denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by \boldsymbol{A}^ or \boldsymbol{A}^{\mathrm{T}{*}} . Example If : \boldsymbol{A} = \begin{bmatrix} 1 & -2-i \\ 1+i & i \end{bmatrix} then : \boldsymbol{A}^{*} = \begin{bmatrix} 1 & 1-i \\ -2+i & -i\end{bmatrix} Basic remarks A square matrix with entries a_{ij} is called * Hermitian or self-adjoint if , i.e. a_{ij}=\overline{a_{ji}} . * skew Hermitian or antihermitian if , i.e. a_{ij}=-\overline{a_{ji}} . * normal if . * unitary if . Even if is not square, the two matrices and are both Hermitian and in fact positive semi-definite matrices. The conjugate transpose "adjoint" matrix should not be confused with the adjugate , which is also sometimes called "adjoint". The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. Motivation The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication: : a + ib \equiv \left(\begin{matrix} a & -b \\ b & a \end{matrix}\right). That is, denoting each complex number by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space \mathbb{R}^2 ) affected by complex -multiplication on \mathbb{C} . An -by- matrix of complex numbers could therefore equally well be represented by a 2 -by-2 matrix of real numbers. The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix, when viewed back again as -by- matrix made up of complex numbers. Properties of the conjugate transpose * for any two matrices and of the same dimensions. * A''∗}} for any complex number and any m''-by-''n matrix . * for any m''-by-''n matrix and any n''-by-''p matrix . Note that the order of the factors is reversed. * for any m''-by-''n matrix . * If is a square matrix, then and . * is invertible if and only if is invertible, and in that case . * The eigenvalues of are the complex conjugates of the eigenvalues of . * ⟨''x, A''∗''y⟩}} for any -by- matrix , any vector in \mathbb{C}^n and any vector in \mathbb{C}^m . Here, \langle\cdot,\cdot\rangle denotes the standard complex inner product on \mathbb{C}^m and \mathbb{C}^n . Generalizations The last property given above shows that if one views as a linear transformation from Euclidean Hilbert space to , then the matrix corresponds to the adjoint operator of . The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis. Another generalization is available: suppose is a linear map from a complex vector space to another, , then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of to be the complex conjugate of the transpose of . It maps the conjugate dual of to the conjugate dual of . Υποσημειώσεις Εσωτερική Αρθρογραφία * Hermitian adjoint * Adjugate matrix * Προσαρτημένη Αναπαράσταση * ελαττωσιμότητα (reducibility) * ανελαττωσιμότητα (Irreducibility) * ορίζουσα * τελεστής * Μήτρα Pauli * Ενιαία Μήτρα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Μαθηματικές Μήτρες